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Physical review. B, Condensed matter

Comment on "Reconciliation of high-temperature series and renormalization-group results by suppressing confluent singularities"

Bernie G. Nickel, Mark Dixon

Published Oct 1, 1982• 21 citations
Publisher

Abstract

We have used a slightly modified version of Roskies's quadratic map method [Phys. Rev. B 24, 5305 (1981)] to estimate the confluent correction amplitudes for the susceptibility $\ensuremath{\chi}$ and correlation length $\ensuremath{\xi}$ of a model that interpolates between Ising and Gaussian limits. Our best estimates for the corresponding leading exponents are $\ensuremath{\gamma}=1.237\ifmmode\pm\else\textpm\fi{}0.003$, $\ensuremath{\nu}=0.630\ifmmode\pm\else\textpm\fi{}0.003$, and $\ensuremath{\eta}=2\ensuremath{-}\frac{\ensuremath{\gamma}}{\ensuremath{\nu}}=0.036\ifmmode\pm\else\textpm\fi{}0.002$. The correction to scaling amplitude ratio $\frac{B(\ensuremath{\xi})}{B(\ensuremath{\chi})}=0.8\ifmmode\pm\else\textpm\fi{}0.1$ is slightly larger than that predicted from ${\ensuremath{\varphi}}^{4}$ continuum model analysis [M.-c. Chang and A. Houghton, Phys. Rev. Lett. 44, 785 (1980) and C. Bagnuls and C. Bervillier, Phys. Rev. B 24, 1226 (1981)]

DOI
10.1103/physrevb.26.3965

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arXiv:hep-lat/0006009•Physical review. B, Condensed matter•Dec 1, 2000•40 citations
We have used a slightly modified version of Roskies's quadratic map method [Phys. Rev. B 24, 5305 (1981)] to estimate the confluent correction amplitudes for the susceptibility \ensuremathχ\ensuremath{\chi}\ensuremathχ and correlation length \ensuremathξ\ensuremath{\xi}\ensuremathξ of a model that interpolates between Ising and Gaussian limits. Our best estimates for the corresponding leading exponents are \ensuremathγ=1.237\ifmmode±\else\textpm\fi0.003\ensuremath{\gamma}=1.237\ifmmode\pm\else\textpm\fi{}0.003\ensuremathγ=1.237\ifmmode±\else\textpm\fi0.003, \ensuremathν=0.630\ifmmode±\else\textpm\fi0.003\ensuremath{\nu}=0.630\ifmmode\pm\else\textpm\fi{}0.003\ensuremathν=0.630\ifmmode±\else\textpm\fi0.003, and \ensuremathη=2\ensuremath−\ensuremathγ\ensuremathν=0.036\ifmmode±\else\textpm\fi0.002\ensuremath{\eta}=2\ensuremath{-}\frac{\ensuremath{\gamma}}{\ensuremath{\nu}}=0.036\ifmmode\pm\else\textpm\fi{}0.002\ensuremathη=2\ensuremath−\ensuremathν\ensuremathγ​=0.036\ifmmode±\else\textpm\fi0.002. The correction to scaling amplitude ratio B(\ensuremathξ)B(\ensuremathχ)=0.8\ifmmode±\else\textpm\fi0.1\frac{B(\ensuremath{\xi})}{B(\ensuremath{\chi})}=0.8\ifmmode\pm\else\textpm\fi{}0.1B(\ensuremathχ)B(\ensuremathξ)​=0.8\ifmmode±\else\textpm\fi0.1 is slightly larger than that predicted from \ensuremathφ4{\ensuremath{\varphi}}^{4}\ensuremathφ4 continuum model analysis [M.-c. Chang and A. Houghton, Phys. Rev. Lett. 44, 785 (1980) and C. Bagnuls and C. Bervillier, Phys. Rev. B 24, 1226 (1981)]
We have extended the series for the zero-field susceptibility of the spin-sss Ising model through tenth order in the reduced inverse temperature KKK on the square, triangular, simple-cubic, body-centered-cubic, and face-centered-cubic lattices. The series coefficients hn(s){h}_{n}(s)hn​(s) are expressed as simple polynomials in X=s(s+1)X=s(s+1)X=s(s+1). Using extended methods of analysis we have estimated the nature of the leading singularities on the face-centered lattice and conclude with good confidence that the susceptibility exponent ${\ensuremath{\gamma}}_{1}$ equals 1.25, independent of sss. The exponent of the leading correction term is estimated to be ${\ensuremath{\gamma}}_{2}\ensuremath{\simeq}0.75\ifmmode\pm\else\textpm\fi{}0.08$ in good agreement with renormalization-group theory. For s=12s=\frac{1}{2}s=21​ only, the amplitude of the confluent correction apparently vanishes. We have also studied the leading singularities on the triangular lattice and conclude that ${\ensuremath{\gamma}}_{1}=\frac{7}{4}$, independent of sss, with, however, much stronger corrections than in three dimensions. These results provide a very strong corroboration of the universality hypothesis.
The susceptibility of the spin-sss nearest-neighbor fcc Ising model is studied at zero magnetic field by twelve-term high-temperature series. We find that the data near criticality are consistent with $\ensuremath{\chi}=A(s){t}^{\frac{\ensuremath{-}5}{4}}[1+B(s){t}^{{\ensuremath{\Delta}}_{1}}]$, $t=1\ensuremath{-}\frac{{T}_{c}(s)}{T}$. The correction-to-scaling exponent has the universal value ${\ensuremath{\Delta}}_{1}=0.50\ifmmode\pm\else\textpm\fi{}0.05$ in good agreement with $\ensuremath{\epsilon}$-expansion estimates. The confluent amplitude B(s)B(s)B(s) vanishes at s=12s=\frac{1}{2}s=21​ to within uncertainties. Values of the critical parameters are tabulated for a variety of spins.
Using existing series for spin-sss Ising models and for continuous-spin Ising models, we have verified the universality of the ratio of the amplitudes associated with the leading confluent corrections to the susceptibility and to the correlation length. We have, thereby, tested the universality of the subcritical part of the correlation function, of which the universality of the amplitude ratio in question is a direct consequence.
Using a renormalized linked-cluster-expansion method, we have extended to order ${\ensuremath{\beta}}^{23}$ the high-temperature series for the susceptibility $\ensuremath{\chi}$ and the second-moment correlation length $\ensuremath{\xi}$ of the spin-1/2 Ising models on the sc and the bcc lattices. A study of these expansions yields updated direct estimates of universal parameters, such as exponents and amplitude ratios, which characterize the critical behavior of $\ensuremath{\chi}$ and $\ensuremath{\xi}.$ Our best estimates for the inverse critical temperatures are ${\ensuremath{\beta}}_{c}^{\mathrm{sc}}=0.221654(1)$ and ${\ensuremath{\beta}}_{c}^{\mathrm{bcc}}=0.1573725(6).$ For the susceptibility exponent we get $\ensuremath{\gamma}=1.2375(6)$ and for the correlation length exponent $\ensuremath{\nu}=0.6302(4).$ The ratio of the critical amplitudes of $\ensuremath{\chi}$ above and below the critical temperature is estimated to be ${C}_{+}{/C}_{\ensuremath{-}}=4.762(8).$ The analogous ratio for $\ensuremath{\xi}$ is estimated to be ${f}_{+}{/f}_{\ensuremath{-}}=1.963(8).$ For the correction-to-scaling amplitude ratio we obtain ${a}_{\ensuremath{\xi}}^{+}{/a}_{\ensuremath{\chi}}^{+}=0.87(6).$